Theorem cotr3 15002

Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)

Hypotheses

Ref	Expression
cotr3.a	⊢ 𝐴 = dom 𝑅
cotr3.b	⊢ 𝐵 = (𝐴 ∩ 𝐶)
cotr3.c	⊢ 𝐶 = ran 𝑅

Assertion

Ref	Expression
cotr3	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3

Step	Hyp	Ref	Expression
1		cotr3.a	. . 3 ⊢ 𝐴 = dom 𝑅
2	1	eqimss2i 4025	. 2 ⊢ dom 𝑅 ⊆ 𝐴
3		cotr3.b	. . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶)
4		cotr3.c	. . . . 5 ⊢ 𝐶 = ran 𝑅
5	1, 4	ineq12i 4198	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅)
6	3, 5	eqtri 2759	. . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅)
7	6	eqimss2i 4025	. 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
8	4	eqimss2i 4025	. 2 ⊢ ran 𝑅 ⊆ 𝐶
9	2, 7, 8	cotr2 15001	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∀wral 3052   ∩ cin 3930   ⊆ wss 3931   class class class wbr 5124  dom cdm 5659  ran crn 5660   ∘ ccom 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670

This theorem is referenced by: (None)